Inner products on a Green ring for finite groups with a cyclic p-Sylow subgroup (Q1057357)

Let G be a finite group with a cyclic p-Sylow subgroup and let R be an unramified extension of the p-adic integers, for some prime number p. Denote by \(p\) the radical of R and by K the field of quotients. Assume that R/\(p=k\) is a splitting field for G, and let L be either R or k. Let \(\alpha_ L(G)\) be the Green ring of the finitely generated L-free LG- modules, and denote by \(\alpha^ 0_ L(G)\) the subring generated by the finitely generated projective LG-modules and the syzygies of the trivial LG-module. Consider the bilinear form [, ]\(=\dim_ kP(, )\) on \(\alpha^ 0_ k(G)\), where P(M,N) denotes the maps from A to B factoring through a projective module. On \(\alpha^ 0_ R(G)\) we consider the form [, ] defined by \([M,N]=\dim_ k(P(M,N)+pHom_{RG}(M,N))/pHom_{RG}(M,N).\) The main result is that unless \(L=R\) and the p-Sylow subgroup of G has order 2, [, ] is nondegenerate on \(\alpha^ 0_ L(G)\). The result for \(L=R\) is proved by reducing to the case \(L=k.\) The form [, ] on \(\alpha^ 0_ L(G)\) is not invariant under stable equivalence, but to prove the result a related form is considered, which is invariant under stable equivalence, even though it is defined in terms of maps factoring through projectives. A connection between Bäckström orders and Brauer trees is also established.